Bounded and Monotonic Sequences Convergence

Given a sequence, determine if it is both bounded and monotonic. If it is, prove that it converges.

In real analysis, understanding the behavior of sequences is a foundational skill. A sequence is a collection of numbers listed in order, and investigating their properties like boundedness and monotonicity is crucial. A sequence is bounded if there exists a real number such that every term in the sequence is less than or equal to this number. Monotonicity refers to whether a sequence consistently increases or decreases. Proving that a sequence is both bounded and monotonic is a strong indication that the sequence converges, meaning it approaches a specific value as the sequence progresses.

Convergence is a cornerstone concept in analysis, particularly when establishing more intricate results about functions and series. When examining whether a sequence converges, it is essential to first check for boundedness and monotonicity, as these conditions significantly simplify the analysis through powerful theorems such as the Monotone Convergence Theorem. This theorem essentially states that every bounded and monotonic sequence converges. Therefore, understanding when these properties apply not only aids in solving the problem at hand but also enhances your ability to tackle more complex problems involving sequences and their limits.